Show that if $x_n$ and $y_n$ are Cauchy sequences then both $x_n+y_n$ and $x_n y_n$ are Cauchy sequences
Show that if $x_n$ and $y_n$ are Cauchy sequences then both $x_n+y_n$ and $x_n y_n$ are Cauchy sequences
The first is simple noting $\left| {(x_m + y_m ) - (x_n + y_n )} \right| \leqslant \left| {x_m - x_n } \right| + \left| {y_m - y_n } \right|$.
$\left| {(x_m y_m ) - (x_n y_n )} \right| \leqslant \left| {(x_m y_m ) - (x_n y_m )} \right| + \left| {(x_n y_m ) - (x_n y_n )} \right|$ $\leqslant \left| {y_m } \right|\left| {x_m - x_n } \right| + \left| {x_n } \right|\left| {y_m - y_n } \right|$